Since it is already bumped, I was wondering if via possible reformulation, the question can be reopened? Many young students like me would benefit immensely; although this is a research level site, the fact is to explain the concepts in expository terms, there is no other better forum than this one with experts.
I am willing to change the question to: Examples of applications of foundational theorems
(1 each, CW) with OP reading:
Formal definition of Goodstein's theorem: "Every Goodstein sequence eventually terminates at 0. The Goodstein sequence G(m) of a number m is a sequence of natural numbers. The first element in the sequence G(m) is m itself. To get the next element, write m in hereditary base 2 notation, change all the 2s to 3s, and then subtract 1 from the result; this is the second element of G(m). To get the third element of G(m), write the second element in hereditary base 3 notation, change all 3s to 4s, and subtract 1 again. Continue until the result is zero, at which point the sequence terminates."
Example by analogy: "Laurie Kirby and Jeff Paris gave an interpretation of the Goodstein's theorem as a hydra game: the "Hydra" is a rooted tree, and a move consists of cutting off one of its "heads" (a branch of the tree), to which the hydra responds by growing a finite number of new heads according to certain rules. The Kirby–Paris interpretation of the theorem says that the Hydra will eventually be killed, regardless of the strategy that Hercules uses to chop off its heads, though this may take a very, very long time."